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# On the classical Gauss sum and the recursive properties

*Advances in Difference Equations*
**volume 2018**, Article number: 387 (2018)

## Abstract

Let *p* be a prime with \(p\equiv1\bmod8\), *ψ* be an eighth character mod*p*, and \(\tau(\psi)\) denote the classical Gauss sum mod*p*. The main purpose of this paper is using the analytic method and the properties of the classical Gauss sum to study the computational problem of one kind rational polynomial of \(\tau(\psi)\). In the end, we prove an interesting second-order linear recursive formula for it.

## Introduction

For any positive integer \(q\geq2\) and Dirichlet character \(\chi\bmod q\), the classical Gauss sum \(\tau(\chi)\) is defined as

where \(e(y) = e^{2\pi i y}\).

This sum is very important in the study of the analytic number theory, so many authors had studied its elementary properties, and obtained a series of important results. In fact, if *χ* is a primitive character mod*q*, then one has the identity \(|\tau(\chi)|=\sqrt{q}\). For the general character \(\chi\bmod q\), we have the estimate \(|\tau (\chi)|\leq\sqrt{q}\). From some special characters \(\chi\bmod q\), the Gauss sum \(\tau(\chi)\) has some interesting properties. For example, if *p* is a prime with \(p\equiv1\bmod3\), and *ψ* is any third-order character \(\bmod\ p\), then we have the identity [1, 2]

where *d* is uniquely determined by \(4p=d^{2}+27b^{2}\) and \(d\equiv1\bmod 3\).

Zhuoyu Chen and Wenpeng Zhang [3] obtained a similar formula for the quartic Gauss sum (see Lemma 2 below). Of course, there are also many similar results, and we are not going to list them here. The reader can refer to Refs. [4–7] and [8] for details.

In this paper, we are considering such a sequence \(F_{k}(p)\) as follows: Let *p* be a prime with \(p\equiv1\bmod8\), *ψ* be any eighth-order character mod*p*. For any integer \(k\geq0\), we define the sequence \(F_{k}(p)\) as

The sequence \(F_{k}(p)\) defined in (1) is clearly a second-order linear recurrence sequence. However, to find the exact value of this sequence, noting that \(F_{0}(p)=2\), we must know its first term \(F_{1}(p)\). Generally, \(F_{1}(p)\) is very difficult to calculate. But if *ψ* is an eighth-order character mod*p*, then we can deduce some interesting results. In this paper, we will focus on illustrating this point. That is, we shall prove the following two results.

### Theorem 1

*Let*
*p*
*be a prime with*
\(p\equiv1\bmod8\), *then*, *for any integer*
\(k\geq2\), *we have the second*-*order linear recursive formula*

*where*
\(F_{0}(p)=2\), \(C=F_{1}(p)=\pm\sqrt{2+\frac{2\alpha}{\sqrt{p}}}\), *and the constant*
\(\alpha=\alpha(p)\)
*is an integer*, *which is closely related to prime*
*p*.

In fact, we have a very important square-sum formula,

where \((\frac{*}{p} )\) denotes the Legendre symbol mod*p*, *r* is any integer with \((\frac{r}{p} )=-1\) (see Theorem 4-11 in [9]).

### Theorem 2

*Let*
*p*
*be a prime with*
\(p\equiv1\bmod8\), *then*, *for any real number*
\(k\geq2\), *we have the second*-*order linear recursive formula*

*where*
\(F_{0}(p)=2\)
*and*
\(F_{2}(p)=\frac{2\alpha}{\sqrt{p}}\).

Since \(|\alpha|\leq\sqrt{p}\), the two roots of the equation \(x^{2}-\frac {2\alpha}{\sqrt{p}}x+1=0\) are

where *i* is the imaginary unit. That is, \(i^{2}=-1\).

Therefore, from the properties of the second-order linear recursive sequence we have the computational formula

*Some notes*: How to determine the positive or negative signs of \(C=F_{1}(p)\) in Theorem 1 is an interesting open problem.

It is clear that \(\phi(5)=\phi(8)=\phi(12)=4\). Therefore, whether there is a similar second-order linear recurrence formula for the fifth-order (or twelfth-order) character mod*p* remains to be further studied.

## Several lemmas

In this section, we need to prove several simple lemmas, which is necessary in the proofs of our theorems. Hereinafter, we shall use many properties of the classical Gauss sum and Dirichlet characters mod*p* (an odd prime); all of them can be found in Ref. [10], so they will not be repeated here.

### Lemma 1

*Let*
*p*
*be a prime with*
\(p\equiv1\bmod8\), *ψ*
*be an eighth*-*order character* mod*p*. *Then we have the identity*

### Proof

Since \(p\equiv1\bmod8\), there exist 4 eighth-order characters mod*p*, let *ψ* be one of them; \(\chi_{2}\) denotes the Legendre symbol mod*p*. For any integer *m* with \((m, p)=1\), note that we have the identities \(\overline{\psi}\chi_{2}=\psi^{3}\) and

from the properties of the classical Gauss sum we have

On the other hand, from the properties of the classical Gauss sum we also have

Note that \(\psi^{4}=\chi_{2}\) and \(\chi_{2}(2)=1\), from (2) and (3) we have the identity

This proves Lemma 1. □

### Lemma 2

*Let*
*p*
*be an odd prime with*
\(p\equiv1\bmod4\), *λ*
*be any fourth*-*order character* mod*p*. *Then we have the identity*

### Proof

See Lemma 2.2 in [3]. □

### Lemma 3

*Let*
*p*
*be a prime with*
\(p\equiv1 \bmod 8\), *ψ*
*be an eighth*-*order character* mod*p*. *Then we have the identity*

*where*
*α*
*is defined as in Theorem *1.

### Proof

It it clear that if *ψ* is an eighth-order character mod*p*, then \(\psi^{3}\) is also an eighth-order character mod*p*. So substituting *ψ* by \(\psi^{3}\) in Lemma 1, and noting that \(\psi ^{9}=\psi\), \(\overline{\psi}^{6}=\psi^{2}\), from Lemma 1 we have

Note that \(\psi^{2}\) is a fourth-order character mod*p*, from (4), Lemma 1 and Lemma 2 we may immediately deduce the identity

which implies the identity

This proves Lemma 3. □

## Proofs of the theorems

Now we will complete the proofs of our main results. First we prove Theorem 1. Let *p* be a prime with \(p\equiv1\bmod8\), *ψ* be an eighth-order character mod*p*. For any integer \(k\geq0\), we define

It is clear that from Lemma 3 we have \(F_{0}(p)=2\) and

From (5) we can deduce that

If \(k\geq1\), then from the definition of \(F_{k}(p)\) we have

or the second-order linear recursive formula

where \(F_{0}(p)=2\) and \(F_{1}(p)=\pm\sqrt{2 (1+\frac{\alpha}{\sqrt {p}} )}\).

This proves Theorem 1.

Similarly, we can deduce Theorem 2. In fact, for any integer \(k\geq1\), from Lemma 3 we have

or the second-order linear recursive formula

where the first two terms are \(F_{0}(p)=2\) and \(F_{2}(p)=\frac{2\alpha }{\sqrt{p}}\).

This completes the proof of Theorem 2.

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## Acknowledgements

The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.

## Funding

This work was supported by the N. S. F. (2017MS0114) of Inner Mongolia and N. S. F. (11771351) of P.R. China.

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### Cite this article

Bai, H., Hu, J. On the classical Gauss sum and the recursive properties.
*Adv Differ Equ* **2018**, 387 (2018). https://doi.org/10.1186/s13662-018-1804-7

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DOI: https://doi.org/10.1186/s13662-018-1804-7

### MSC

- 11L05

### Keywords

- The classical Gauss sum
- Eighth character
- Rational polynomial
- Second-order linear recursive formula
- Analytic method